2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143937We reformulate the integrality property of the Poincaré inner product in the middle dimension, for an arbitrary Poincaré $\Q$-algebra, in classical terms (discriminant and local invariants). When the algebra is 1-connected, we show that this property is the only obstruction to realizing it by a closed manifold, up to dimension 11. We reinterpret a result of Eisenbud and Levine on finite map germs, relating the degree of the map germ to the signature of the associated local ring, to answer a question of Halperin on artinian weighted complete intersections.We analyse the homogeneous artinian complete intersections over $\Q$ realized by closed manifolds of dimensions 4 and 8, and their signatures.13 pagesAlgebraic Topology57R65;13C40;11E81;58K20 58K20 58K20text